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Notes on the Construction of the Moduli Space of Curves

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Notes on the Construction of the Moduli Space of Curves NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES DAN EDIDIN The purpose of these notes is to discuss the problem of moduli for curves of genus g ≥ 3 1 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into 4 parts. In Section 1 we discuss the general problem of constructing a moduli “space” of curves. We will also state results about its properties, some of which will be discussed in the sequel. We begin Section 2 by recalling from [DM] (see also [Vi]) the defini- tion of a groupoid, and define the moduli groupoid of curves, as well as the quotient groupoid of a scheme by a group. We then discuss mor- phisms and fiber products in the 2-category of groupoids. Once this is in place we state the geometric conditions required for a groupoid to be a stack, and prove that the quotient groupoid of a scheme by a group is a stack. After discussing properties of morphisms of stacks, we define a Deligne-Mumford stack and prove that if a group acts on a scheme so that the stabilizers of geometric points are finite and re- duced then the quotient stack is Deligne-Mumford. In the last part of Section 2 we talk about some basic algebro-geometric properties of Deligne-Mumford stacks. In Section 3 the notion of a stable curve is introduced, and we define the groupoid of stable curves. The groupoid of smooth curves is a sub- groupoid. We then prove that the groupoid of stable curves of genus g ≥ 3 is equivalent to the quotient groupoid of a Hilbert scheme by the action of the projective linear group with finite, reduced stabilizers at geometric points. By the results of the previous section we can conclude the the groupoid of stable curves is a Deligne-Mumford stack defined over Spec Z (as is the groupoid of smooth curves). We also discuss the results of [DM] on the irreducibility of the moduli stack. We begin Section 4 by defining the moduli space of a Deligne-Mumford stack, proving that a geometric quotient of a scheme by a group is the The author was partially supported by an N.S.A grant during the preparation of this work. 1Because every curve of genus 1 and 2 has non-trivial automorphisms, the prob- lem of moduli is more subtle in this case than for curves of higher genus 1 moduli space of the quotient stack. We then discuss the method of geo- metric invariant theory for constructing geometric quotients for actions of reductive groups. Finally, we briefly outline Gieseker’s approach to constructing the coarse moduli space over an algebraically closed field as the quotient of Hilbert schemes of pluricanonically embedded stable curves. Acknowledgments: These notes are based on lectures the author gave at the Weizmann Institute, Rehovot, Israel in July 1994. It is a pleasure to thank Amnon Yekutieli and Victor Vinnikov for the invitation, and for many discussions on the material in these notes. Special thanks to Angelo Vistoli for a careful reading and many critical comments. Thanks also to Alessio Corti, Andrew Kresch and Ravi Vakil for useful discussions. 1. Basics Definition 1.1. Let S be a scheme. By a smooth curve of genus g over S we mean a proper, smooth family C → S whose geometric fibers are smooth, connected 1-dimensional schemes of genus g. Remark 1.1. By the genus of a smooth, connected curve C over an 0 1 algebraically closed field, we mean dim H (C, ωC ) = dim H (C, OC ) = g, where ωC is the sheaf of regular 1-forms on C. If the ground field is C, then C is a smooth, compact Riemann surface, and the algebraic definition of genus is the same as the topological one. The basic problem of moduli is to classify curves of genus g. As a start, it is desirable to construct a space Mg whose geometric points represent all possible isomorphism classes of smooth curves. In the lan- guage of complex varieties, we are looking for a space that parametrizes all possible complex structures we can put on a fixed surface of genus g. However, as modern (post-Grothendieck) algebraic geometers we would like Mg to have further functorial properties. In particular given a scheme S, a curve C → S should correspond to a morphism of S to Mg (when S is the spectrum of an algebraically closed field this is exactly the condition of the previous paragraph). In the language of functors, we are trying to find a scheme Mg which represents the functor FMg : Schemes → Sets which assigns to a scheme S the set of isomorphism classes of smooth curves of genus g over S. Unfortunately, such a moduli space can not exist because some curves have non-trivial automorphisms. As a result, it is possible to construct non-trivial families C → B where each fiber has the same isomorphism class. Since the image of B under the corresponding map to the moduli 2 space is a point, if the moduli space represented the functor FMg then C → B would be isomorphic to the trivial product family. Given a curve X and a non-trivial (finite) group G of automorphisms of X we construct a non-constant family C → B where each fiber is isomorphic to X as follows. Let B0 be a scheme with a free G action, and let B = B0/G be the quotient. Let C0 = B0 × X. Then G acts on C0 by acting as it does on B0 on the first factor, and by automorphism on the second factor. The quotient C0/G is a family of curves over B. Each fiber is still isomorphic to X, but C will not in general be isomorphic to X × B. There is however, a coarse moduli scheme of smooth curves. By this we mean: Definition 1.2. ([GIT, Definition 5.6, p.99]) There is a scheme Mg and a natural transformation of functors φ : FMg → Hom(·, Mg) such that 1. For any algebraically closed field k, the map φ : FMg (Ω) → Hom(Ω, Mg) is a bijection, where Ω = Spec k. 2. Given any scheme M and a transformation ψ : FMg → Hom(·,M) there is a unique transformation χ : Hom(·, Mg) → Hom(·,M) such that ψ = χ ◦ φ. The existence of φ means that given a family of curves C → B there is an induced map to Mg. We do not require however, that a map to moduli gives a family of curves (as we have already seen a non-constant family with iso-trivial fibers). However, condition 1 says that giving a curve over an algebraically closed field is equivalent to giving a map of that field into Mg. Condition 2 is imposed so that the moduli space is a universal object. In his book on geometric invariant theory Mumford proved the fol- lowing theorem. Theorem 1.1. ([GIT, Chapter 5]) Given an algebraically closed field k there is a coarse moduli scheme Mg of dimension 3g −3 defined over Spec k, which is quasi-projective and irreducible. The proof of this theorem will be subsumed in our general discussion of the construction of Mg, the moduli space of stable curves. A natural question to ask at this point, is whether Mg is complete (and thus projective). The answer is no. It is quite easy to construct curves C → Spec O with O a D.V.R. which has function field K, where C is smooth, but the fiber over the residue field is singular. Since C is smooth, the restriction CK → Spec K is a smooth curve over Spec K, so there is a map Spec K → Mg. The existence of such a family does 3 not prove anything, since we must show that we can not replace the special fiber by a smooth curve. The total space C˜ of a family with modified special fiber is birational to C. Since C and C˜ are surfaces (being curves over 1-dimensional rings) there must be a sequence of birational transformations (centered in the special fiber) taking one to the other. It appears therefore, that it suffices to construct a family C → Spec O such that no birational modification of C centered in the special fiber will make it smooth. Unfortunately, because Mg is only a coarse moduli scheme, the existence of such a family does not prove that Mg is incomplete. The reason is that there may be a map Spec O → Mg extending the original map Spec K → Mg without ˜ there being a family of smooth curves C → Spec O extending CK → Spec K. However, we will see when we discuss the valuative criterion of properness for Deligne-Mumford stacks that it suffices to show that for every finite extension K ⊂ K0 we can not complete the induced 0 family of smooth curves CK0 → Spec K to a family of smooth curves C0 → Spec O0, where O0 is the integral closure of O in K0. Example 1.1. The following family shows that M3 is not complete. It can be easily generalized to higher genera. Consider the family x4 + xyz2 + y4 + t(z4 + z3x + z3y + z2y2) of quartics in P2 × Spec O where O is a D.V.R. with uniformizing parameter t. The total space of this family is smooth, but over the closed point the fiber is a quartic with a node at the point (0 : 0 : 1) ∈ P2. Moreover, even after base change, any blow-up centered at the singular point of the special fiber contains a (−2) curve, so there is no modification that gives us a family of smooth curves.
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